Parallel Quantum Algorithm for Hamiltonian Simulation

Zhicheng Zhang1,2, Qisheng Wang3,4, and Mingsheng Ying5,4

1Centre for Quantum Software and Information, University of Technology Sydney, Sydney, Australia
2University of Chinese Academy of Sciences, Beijing, China
3Graduate School of Mathematics, Nagoya University, Nagoya, Japan
4Department of Computer Science and Technology, Tsinghua University, Beijing, China
5State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, China

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We study how parallelism can speed up quantum simulation. A parallel quantum algorithm is proposed for simulating the dynamics of a large class of Hamiltonians with good sparse structures, called uniform-structured Hamiltonians, including various Hamiltonians of practical interest like local Hamiltonians and Pauli sums. Given the oracle access to the target sparse Hamiltonian, in both query and gate complexity, the running time of our parallel quantum simulation algorithm measured by the quantum circuit depth has a doubly (poly-)logarithmic dependence $\operatorname{polylog}\log(1/\epsilon)$ on the simulation precision $\epsilon$. This presents an $\textit{exponential improvement}$ over the dependence $\operatorname{polylog}(1/\epsilon)$ of previous optimal sparse Hamiltonian simulation algorithm without parallelism. To obtain this result, we introduce a novel notion of parallel quantum walk, based on Childs' quantum walk. The target evolution unitary is approximated by a truncated Taylor series, which is obtained by combining these quantum walks in a parallel way. A lower bound $\Omega(\log \log (1/\epsilon))$ is established, showing that the $\epsilon$-dependence of the gate depth achieved in this work cannot be significantly improved.
Our algorithm is applied to simulating three physical models: the Heisenberg model, the Sachdev-Ye-Kitaev model and a quantum chemistry model in second quantization. By explicitly calculating the gate complexity for implementing the oracles, we show that on all these models, the total gate depth of our algorithm has a $\operatorname{polylog}\log(1/\epsilon)$ dependence in the parallel setting.

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[2] Kouhei Nakaji, Shumpei Uno, Yohichi Suzuki, Rudy Raymond, Tamiya Onodera, Tomoki Tanaka, Hiroyuki Tezuka, Naoki Mitsuda, and Naoki Yamamoto, "Approximate amplitude encoding in shallow parameterized quantum circuits and its application to financial market indicators", Physical Review Research 4 2, 023136 (2022).

[3] Pei Yuan and Shengyu Zhang, "Optimal (controlled) quantum state preparation and improved unitary synthesis by quantum circuits with any number of ancillary qubits", Quantum 7, 956 (2023).

[4] Qisheng Wang and Zhicheng Zhang, "Fast Quantum Algorithms for Trace Distance Estimation", arXiv:2301.06783, (2023).

[5] John M. Martyn, Yuan Liu, Zachary E. Chin, and Isaac L. Chuang, "Efficient Fully-Coherent Quantum Signal Processing Algorithms for Real-Time Dynamics Simulation", arXiv:2110.11327, (2021).

[6] Xiao-Ming Zhang and Xiao Yuan, "On circuit complexity of quantum access models for encoding classical data", arXiv:2311.11365, (2023).

[7] Nai-Hui Chia, Kai-Min Chung, Yao-Ching Hsieh, Han-Hsuan Lin, Yao-Ting Lin, and Yu-Ching Shen, "On the Impossibility of General Parallel Fast-forwarding of Hamiltonian Simulation", arXiv:2305.12444, (2023).

[8] Gregory Boyd, "Low-Overhead Parallelisation of LCU via Commuting Operators", arXiv:2312.00696, (2023).

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